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We present a formulation of measurement-based feedback control of a singlequantum particle in one spatial dimension. An arbitrary linear combination ofthe position and momentum of the particle is continuously monitored, andfeedback proportional to the measured signal is used to control the system. Wederive a feedback master equation and discuss a general approach to computingthe steady-state solutions for arbitrary potentials. For a quantum harmonicoscillator or a free particle, we show that it is possible to cool and confinethe system using feedback that simultaneously damps the measured observable andits conjugate momentum, as well as compensates for noise introduced by themeasurement. In addition, we demonstrate that appropriate feedback adds aquadratic term in the measured observable to the Hamiltonian of the system. Forthe particular case of the harmonic potential, we describe the exact dynamicsof the system that can be cooled to the ground state. Moreover, we provide anargument for the possibility to cool systems with arbitrary potentials providedthat the measurement is strong enough to localise the particle on an intervalsmaller than the characteristic length scale of the potential.

Measurement-based Feedback Control of a Quantum System in a Harmonic Potential